Seismic exploration involves surveying subterranean geological media for hydrocarbon deposits and estimating various physical properties of the geological formations. Some surveys are known as “marine” surveys because they are conducted in marine environments. However, “marine” surveys may be conducted not only in saltwater environments, but also in fresh and brackish waters. In one type of marine survey, called a “towed-array” survey, an array of seismic sensor-containing streamers and sources is towed behind a survey vessel.
A survey typically involves deploying seismic source(s) and seismic sensor(s) at predetermined locations. The sources generate seismic waves, which propagate into a geological medium creating pressure changes and vibrations along their way. Variations in physical properties of the geological medium change the seismic waves, such as their direction of propagation and other properties. Parts of the seismic waves reach the seismic sensors. Some seismic sensors are sensitive to pressure changes (hydrophones), others to particle motion (e.g., geophones), and industrial surveys may deploy only one type of sensors or both. In response to the detected seismic waves, the sensors generate corresponding electrical signals and record them in storage media as seismic data.
Analysis of the seismic data can be performed to process the seismic data into an image of the geological medium. This image and additional seismic attributes are used to estimate the physical properties of the geological formations and their hydrocarbon bearing potential. Reverse-time migration (RTM) propagates wavefields at the source locations into the geological medium forward in time and recorded wavefields at the receiver locations into the geological medium backward in time and then correlates the two types of wavefields to form an image of the geological medium. Full waveform inversion (FWI) goes a step further and similarly to RTM produces an image of the geological medium but additionally it requires that the simulated wavefields from this image closely match the acquired data. This is typically accomplished by using iterative inversion techniques. To reduce the computational cost of RTM and FWI while still allowing anisotropy, pseudoacoustic systems of differential equations are constructed that are less computationally expensive than using a fully elastic system. Some conventional approaches modify the dispersion relation for either the fully elastic system or a pseudoacoustic approximation (e.g., a 2×2 second-order pseudoacoustic systems of differential equations) by setting an S-wave velocity to be zero, and eliminating one or more corresponding terms in the pseudoacoustic system of differential equations. Unfortunately, because physical conservation laws may not be satisfied under such approximations, doing so can result in the introduction of instabilities. These instabilities arise from stationary noise that grows with time, thus making imaging of deeper layers or complex geological structures in the geological medium difficult. Moreover, it has been found that these instabilities can persist even when the S-wave velocity is artificially set to be greater than zero. This is an even larger problem for FWI, for two reasons. Firstly, it is typically an iterative procedure and any instability is significantly amplified during the iterations. Secondly, inversion methods such as FWI require that the backward and forward wavefields are propagated via adjoint propagator methods to satisfy the requirements of the inversion algorithms. This requirement is not currently satisfied in FWI applications for tilted transverse isotropy (TTI) medium and approximate and ad hoc forward and backward propagator pairs are used greatly reducing fidelity and efficiency.